Quantum state tomography (QST), the process of reconstructing some unknown quantum state $\hat\rho$ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error $\Delta\hat\rho = \hat\rho-\hat\rho^\prime$ in a QST reconstruction $\hat\rho^\prime$ is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors $\Delta\hat\rho$ and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of $\Delta\hat\rho$ can be evaluated by substituting samples of an appropriate GUE for realizations of $\Delta\hat\rho$. This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance $\|\Delta\hat\rho\|_\mathrm{Tr}$ (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future.

翻译：量子态层析（QST）是通过对未知量子态 $\hat\rho$ 的多个副本进行重复测量以重构该态的过程，在量子计算与量子模拟领域具有基础性的重要意义。因此，对 QST 重构结果 $\hat\rho^\prime$ 中的误差 $\Delta\hat\rho = \hat\rho-\hat\rho^\prime$ 进行详细刻画，对于量子理论与实验显然至关重要。在本工作中，我们针对信息完备基下的态层析发展了一套完整的随机矩阵理论（RMT）处理方法；并在此过程中揭示了 QST 误差 $\Delta\hat\rho$ 与高斯幺正系综（GUE）之间的深刻联系。通过利用这一联系，我们证明了 $\Delta\hat\rho$ 谱的广泛函数类可以通过用适当 GUE 的样本替代 $\Delta\hat\rho$ 的实现来进行计算。这一强大而灵活的结果使得能够对以迹范数 $\|\Delta\hat\rho\|_\mathrm{Tr}$ 量化的误差的均值与方差进行简洁的解析处理（我们针对常见层析协议进行了数值验证），使我们能够推导出 QST 样本复杂度的界限，并随后证明该界限在最广泛使用的再物理化程序下保持不变。这些结果共同展示了我们方法的灵活性、强度与广泛适用性，并为未来更广泛的 QST 随机矩阵理论研究奠定了基础。